The contravariant 4-displacement is:

$${x}^{\alpha} = (ct,\mathbf{r})$$

And the contravariant 4-gradient is:

$${\partial}^{\alpha} = (\frac{1}{c}\frac{\partial}{\partial{t}},-\nabla)$$

From what I can gather so far, other contravariant 4-vectors tend to follow the same pattern as for the displacement - only with the 4-gradient does the negative spatial part appear in the contravariant form. Can someone please explain why the negative is in this one, and not the covariant 4-gradient instead?

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    $\begingroup$ What is the signature convention used for $\eta_{\alpha\beta}$? $\endgroup$ – Everiana Mar 13 '16 at 0:08
  • $\begingroup$ Just because in the presence of a metric, we can convert vectors into dual vectors (or "contravariant vectors" into "covariant vectors", if you insist) and vice versa, some objects have more natural meaning in certain "variance" than others. For example the electromagnetic tensor $F_{\mu\nu}$ is naturally covariant. The flat space derivative operator $\partial_\mu$ naturally exists like this, even without a metric, so the "contravariant" derivative $\partial^\mu$ is the one that will have "unnatural" signs. $\endgroup$ – Bence Racskó Mar 13 '16 at 18:54

To give you a short and direct answer. The gradient is naturally a covariant object and it seems that in the convention of the text you are reading, you are working in a Minkowski space-time (flat space-time), with a metric g (+,-,-,-). Im sure you've heard of the "Index gymnastics" that the metric is used for, as taught in most classes.

So by definition, the contravariant "partner" (changing the gradient from covariant --> contravariant) will pick up minus signs in the last 3 components, hence the negative del operator.

To be honest, this is really an issue on convention, owing to the symmetries of Minkowski space-time (ST) and its metric. In Minkowski ST, one can be slightly be loose with their vectors, contra and covariant and get away with it permitted one is consistent, however, in curved spacetime this bad practice will lead to serious difficulties.

I hope this answer was sufficient.


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